Rational to Decimal

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Answer Key & Explanations

Rational to Decimal · Grade 7 · Worksheet 1

1. Liam is tracking his savings account balance. He started with $250.75, then deposited his weekly allowance of $45.50. Later, he withdrew $18.25 to buy a book. What is Liam's new account balance? Answer: 278.00 Solution: Start with the initial balance. Liam started with $250.75. Initial balance = 250.75 Add the weekly allowance deposit.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Start with the initial balance.** Liam started with $250.75. Initial balance = 250.75 --- **Step 2: Add the weekly allowance deposit.** Allowance = 45.50 New balance after deposit = 250.75 + 45.50 Calculate: 250.75 + 45.50 = 296.25 --- **Step 3: Subtract the withdrawal for the book.** Withdrawal = 18.25 New balance after withdrawal = 296.25 - 18.25 Calculate: 296.25 - 18.25 = 278.00 --- **Step 4: Final answer** Liam's new account balance is $278.00.

2. Convert 11/18 to decimal. Answer: 0.61̄ Solution: Divide 11 by 18. 18 goes into 11 zero times, so add a decimal point and a zero: 110. 18 goes into 110 six times (18 × 6 = 108), remainder 2.
   Full step-by-step solution

   Step 1: Divide 11 by 18. 18 goes into 11 zero times, so add a decimal point and a zero: 110. Step 2: 18 goes into 110 six times (18 × 6 = 108), remainder 2. Step 3: Bring down a zero: 20. 18 goes into 20 one time (18 × 1 = 18), remainder 2. Step 4: Notice the remainder is 2 again. Each time we bring down a zero, we get 20, and 18 goes into 20 one time with remainder 2. So the digit '1' will repeat forever. Step 5: The decimal is 0.61111... which is written as 0.61̄ (the bar over the 1 shows it repeats). The answer is 0.61̄.

3. Liam is designing a rectangular garden for his school's science project. The garden's length is 12.75 meters and its width is 8.4 meters. He needs to calculate the total area to determine how much soil to buy. What is the area of the garden in square meters? Answer: 107.1 Solution: To find the area of a rectangle, we multiply the length by the width. Write down the given measurements. Length = 12.75 meters Width = 8.4 meters Set up the multiplication for the area.
   Full step-by-step solution

   To find the area of a rectangle, we multiply the length by the width. Step 1: Write down the given measurements. Length = 12.75 meters Width = 8.4 meters Step 2: Set up the multiplication for the area. Area = Length × Width Area = 12.75 × 8.4 Step 3: To make multiplying decimals easier, we can think of this as multiplying whole numbers and then adjusting for the decimal places later. First, multiply 12.75 by 10 to make it a whole number: 12.75 × 10 = 127.5. It's easier to work with whole numbers, so let's multiply both numbers by powers of 10 to eliminate decimals entirely. 12.75 has two digits after the decimal. 8.4 has one digit after the decimal. This means the product will have 2 + 1 = 3 digits after the decimal. Step 4: Multiply the numbers as if they were whole numbers. 1275 (which is 12.75 × 100) × 84 (which is 8.4 × 10) ------ First, multiply 1275 by 4 (the ones digit of 84): 1275 × 4 = 5100 Next, multiply 1275 by 80 (the tens digit of 84). Since it's actually 80, we can multiply by 8 and then add a zero: 1275 × 8 = 10200 Now add a zero to make it 102000 (because we are multiplying by 80, not 8). Step 5: Add the two partial products together. 5100 + 102000 --------- 107100 So, 1275 × 84 = 107,100. Step 6: Now we must account for the decimals we removed earlier. We originally multiplied 12.75 (two decimal places) and 8.4 (one decimal place). To get back to the original numbers, we need to divide our result by 100 (for 12.75) and by 10 (for 8.4). This is the same as dividing by 100 × 10 = 1000. So, we take our product 107100 and divide it by 1000. Dividing by 1000 moves the decimal point three places to the left. 107100 ÷ 1000 = 107.100, which is 107.1 Step 7: State the final answer with the correct units. Therefore, the area of the garden is 107.1 square meters.

4. Mere is designing a geometric art piece on a coordinate grid. She draws a large square with vertices at (0, 0), (24, 0), (24, 24), and (0, 24). Inside this square, she marks a smaller square whose vertices are at (4, 4), (20, 4), (20, 20), and (4, 20). The region between the large square and the small square is to be filled with a repeating decimal pattern. If the area of the large square is 576 square units and the area of the small square is 256 square units, what is the decimal representation of the fraction of the large square's area that is occupied by the small square? Answer: 0.444... Solution: Find the area of the large square. Side length = 24, so area = 24 x 24 = 576 square units. Find the area of the small square.
   Full step-by-step solution

   Step 1: Find the area of the large square. Side length = 24, so area = 24 x 24 = 576 square units. Step 2: Find the area of the small square. Side length = 20 - 4 = 16, so area = 16 x 16 = 256 square units. Step 3: The fraction of the large square occupied by the small square is 256/576. Step 4: Simplify the fraction. Divide numerator and denominator by 64: 256 ÷ 64 = 4, 576 ÷ 64 = 9, so the fraction is 4/9. Step 5: Convert 4/9 to a decimal by dividing 4 by 9. 9 goes into 4 zero times, remainder 4. Add a decimal point and zero: 40 ÷ 9 = 4, remainder 4. This pattern repeats: 4 ÷ 9 = 0.444... The answer is 0.444... (repeating).

5. Mason is designing a mosaic on a coordinate grid. The mosaic features a large rectangle with vertices at (0, 0), (77, 0), (77, 42), and (0, 42). Inside this rectangle, he draws a diagonal line from (0, 0) to (77, 42). He then colors the region below this diagonal line. What fraction of the rectangle's area is colored? Express your answer as a simplified fraction and as a repeating decimal (use bar notation for the repeating part). Answer: Fraction: 1/2, Decimal: 0.5 Solution: Find the area of the rectangle. The rectangle has length 77 units and width 42 units. Area = length × width = 77 × 42 = 3234 square units.
   Full step-by-step solution

   Step 1: Find the area of the rectangle. The rectangle has length 77 units and width 42 units. Area = length × width = 77 × 42 = 3234 square units. Step 2: Identify the shape below the diagonal. The diagonal line from (0, 0) to (77, 42) divides the rectangle into two triangles. The region below the diagonal is a triangle with base 77 units (along the x-axis) and height 42 units (vertical distance from the x-axis to the top-right corner at y=42). Step 3: Calculate the area of the triangle. Area of a triangle = (1/2) × base × height = (1/2) × 77 × 42 = (1/2) × 3234 = 1617 square units. Step 4: Find the fraction of the rectangle that is colored. Fraction = (area of triangle) / (area of rectangle) = 1617 / 3234. Simplify the fraction: 1617 ÷ 1617 = 1, and 3234 ÷ 1617 = 2. So the simplified fraction is 1/2. Step 5: Convert 1/2 to a decimal. 1/2 = 0.5. This is a terminating decimal, not a repeating one, so it can be written as 0.5 (no bar needed, as 0.5 has no repeating cycle). The answer is Fraction: 1/2, Decimal: 0.5.